Suppose a particular population has two kinds of health risks, high and low. Let the expected annual health care costs for the high risk be $5,000, and for the low risk, half that. If there are twice as many low risk as high risk individuals, and if the one insurer’s administrative load is 20%, what would the premium be if everyone is compelled to and able to buy health insurance?
To calculate the premium, we need to consider a few factors: the expected costs for the high and low-risk individuals, the number of individuals in each group, and the administrative load. Here's how we can calculate the premium:
1. Determine the expected annual health care cost for the low-risk individuals. Given that the expected cost for the high-risk individuals is $5,000, the expected cost for the low-risk individuals would be half that, which is $2,500.
2. Determine the ratio of low-risk to high-risk individuals. The problem states that there are twice as many low-risk individuals as high-risk individuals. So, let's assume there are x high-risk individuals, which means there are 2x low-risk individuals.
3. Calculate the expected total cost for both high and low-risk individuals. Multiplying the number of individuals in each group by their respective expected costs gives us:
Total cost for high-risk individuals = x * $5,000 = $5,000x
Total cost for low-risk individuals = 2x * $2,500 = $5,000x
4. Determine the administrative load, which is given as 20%. This means that the insurer's administrative costs will be 20% of the total expected cost.
Administrative load = 0.20 * (Total cost for high-risk individuals + Total cost for low-risk individuals)
5. Calculate the total expected cost, including the administrative load:
Total expected cost = (Total cost for high-risk individuals + Total cost for low-risk individuals) + Administrative load
6. Finally, calculate the premium:
Premium = Total expected cost / (Total number of high-risk individuals + Total number of low-risk individuals)
By following these steps and plugging in the appropriate values, you can determine the premium when everyone is compelled to buy health insurance.